This section reviews earlier work on imploding waves. For clarity, the literature has been separated into three sections: imploding shock waves, imploding detonation waves, and imploding toroidal waves.
1.3.1
Imploding Cylindrical and Spherical Shock Waves
The imploding shock solution was first solved analytically in a self-similar fashion for cylindrical and spherical geometries by Guderley (1942) and later reworked by others (Butler, 1954, Sedov, 1959, Stanyukovich, 1960, Dyke and Guttmann, 1982, Ponchaut, 2005) seeking to improve upon its accuracy. The solution assumes that the trajectory of the imploding and reflected shocks follow a power law. As the shock radius decreases to zero, the solution becomes singular. An approximate solution, referred to as the Chester-Chisnell-Whitham (CCW) theory, was found for shock- wave propagation in channels with varying cross-sectional areas by Chester (1954) and Chisnell (1955) and independently by Whitham (1958).
Perry and Kantrowitz (1951) first published experimental observations of the high- temperature focal region created by such an implosion. They (Perry and Kantrowitz, 1951) used a shock tube with a teardrop-shaped obstruction to shape a planar shock wave into a cylindrically imploding wave. While they did not obtain pressure mea- surements, they were able to image luminosity emitted from ionized argon at the
Since the work of Perry and Kantrowitz (1951), several similar experiments have been used to further characterize imploding shock waves. The facility of Wu et al. (1980) tested the ability of differently shaped teardrop obstructions to create sym- metrical implosions. Out of the three obstructions tested (a logarithmic spiral, a five-element contraction, and a three-element conical contraction), the three-element conical contraction was found to provide the best performance, and the results agreed well with the CCW theory except when the imploding shock radius became very small. Matsuo and Nakamura (1981) used explosive cylindrical PETN shells to create im- ploding cylindrical shock waves in air. The shock waves created with this technique were of sufficient strength to ionize the air. Wave trajectories were measured with ion- ization probes and were found to agree with Guderley’s work (Guderley, 1942). More recently, an annular vertical shock tube (Hosseini et al., 1998, 2000) was constructed that uses a rubber sheet under pressure to separate the driver gas from the driven gas. Instead of rupturing a diaphragm, the pressure supporting the rubber sheet is relaxed and the sheet retracts, creating an imploding, ring-shaped shock wave in the test section. The facility is intended to study the stability of imploding shock waves. All previous experimental results on imploding shock waves (Perry and Kantrowitz, 1951, Wu et al., 1980, Matsuo and Nakamura, 1981, Takayama et al., 1987) have in- dicated that disturbances in the wave front (due to diaphragm opening or shock tube supports) become amplified as the wave implodes, resulting in growing nonuniformi- ties at small radii. Numerical simulations of imploding shock waves (Sod, 1977, Fong and Ahlborn, 1979, Wang, 1982, Schwendeman, 2002) show growth of disturbances imposed on the boundary or initial conditions of the flow.
1.3.2
Imploding Cylindrical and Spherical Detonation Waves
Zel’dovich (1959) also observed that the implosion process would result in additional compression behind detonation waves, noting that the release of energy from the deto- nation reaction would eventually become negligible compared to the energy imparted
to the flow by shock processing in the final stages of the implosion process. Lee and Lee (1965) generated cylindrically imploding detonation waves in acetylene-oxygen mixtures and used pressure transducers and streak photographs to characterize the implosion process. They also extended the model of Whitham (1958) to detonation waves and found good agreement between their experiment and theory, measuring focal pressures of 18 times PCJ.
Terao (1983), Terao and Wagner (1991), and Terao et al. (1995) have also per- formed studies on imploding detonation waves in spherical and cylindrical geometries with propane-oxygen mixtures and have characterized the imploding wave with ion- ization probes, pressure transducers, and soot foils. In their experiments, it was found that the experimentally measured wave acceleration was lower than that predicted by the theory of Guderley (1942), but that the post-detonation pressures were higher than theory. Terao and Wagner (1991) attributed such differences to the proximity of the experiment walls to the implosion.
A number of researchers have also attempted to measure the temperature near the focus of imploding detonation waves. Knystautas et al. (1969) inferred the tem- perature at the focus of a cylindrically imploding, acetylene-oxygen detonation wave using spectroscopic techniques and used Wein’s Law to estimate that the maximum temperature was on the order of 200,000 K. Subsequent studies by Roberts and Glass (1971) and Roig and Glass (1977) measured focal temperatures of 4,500-6,000 K in hydrogen-oxygen mixtures. They (Roig and Glass, 1977) also suggested that Knys- tautas et al. (1969) may have incorrectly applied Wein’s Law in arriving at such a large focal temperature. Further work by Saito and Glass (1982) with the appara- tus of Roig and Glass (1977) measured peak temperatures of 10,000-13,000 K and also used PETN explosive shells around the periphery of the chamber to boost the measured temperature to 15,000-17,000 K. Matsuo et al. (1985) continued their pre- vious experimental study, taking spectroscopic measurements of the temperature of cylindrically imploding waves in air. They found that the maximum temperature measured was approximately proportional to the square root of the initiation energy and they measured temperatures that ranged from 13,000-34,000 K, depending on the
temperatures that would be estimated from the shock propagation speed alone. Terao et al. (1995) have also used a laser-light scattering method to measure temperatures of 107-108 K at the focus of spherical imploding detonation waves in propane-oxygen mixtures from an initial wave radius of 500 mm.
Simulations of imploding detonation waves by Devore and Oran (1992) and Oran and Devore (1994) observed that, when disturbances were imposed on the wave from a tube support or similar obstacle, the imploding detonation became at least as unstable as shock waves under similar conditions, if not more so. This was in contrast to conclusions inferred from the experimental work of Knystautas and Lee (1971), who determined that imploding detonations were relatively stable.
1.3.3
Imploding Toroidal Waves
While all of the previously mentioned research was concerned with either cylindrically or spherically imploding shock waves, several studies have also been performed with toroidally imploding waves issuing from annular orifices. Simulations by Jiang and Takayama (1998) in air showed that a diffracting toroidal wave discharged from an annular gap created a region of intense shock-focusing when the toroidal waves merged at the axis of symmetry.
Murray et al. (2000) quantified the effectiveness of this geometry on detonation initiation while conducting experiments measuring the transfer of a detonation wave from a smaller-diameter initiator tube to a larger-diameter test-section tube. The initiator tube and test-section tube were both filled with a hydrogen-air mixture, and several different obstacles were placed between the two tubes. The effect of these obstacles on the detonation wave transmission was measured in terms of its transmission efficiency. Values of the transmission efficiency above unity represent situations where the obstacle allowed detonation transfer from the initiator tube to the test-section tube for mixtures with larger cell sizes than in the case where no obstacles were used. Conversely, values of the transmission efficiency below unity
required that smaller cell size mixtures be used (compared to the no-obstacle case) to transfer the detonation wave between the initiator tube and the test-section tube. When using obstacles consisting of a circular plate, Murray et al. (2000) noted a substantial increase in the transmission efficiency. The obstacle created an annular orifice that generated an imploding toroidal shock wave downstream of the obsta- cle, which was trailed by a deflagration. Murray et al. (2000) demonstrated with numerical simulations that the focus of this imploding toroid was a region of high energy density, which was responsible for reinitiation of a self-sustaining detonation wave. Specifically, Murray et al. (2000) determined that the annular orifice allowed successful detonation transmission for tubes with diameters 2.2 times smaller than cases where no obstacles were located at the interface.
Improving on this concept, a detonation initiator has been developed (Jackson and Shepherd, 2002, Jackson et al., 2003) that successfully detonates propane-air mixtures inside a detonation tube using an imploding toroidal wave propagated into the propane-air mixture from the tube walls. In order to generate the imploding wave, the toroidal initiator uses a single spark plug and a small amount of acetylene-oxygen gas.
Recent numerical simulations (Li and Kailasanath, 2003b, 2005) proposed to im- prove upon this concept by using an imploding toroidal shock wave (instead of an imploding detonation wave) driven by jets of air or fuel. In particular, Li and Kailasanath (2005) computed that an imploding annular jet with a Mach number of unity, a pressure of 2.0 bar, and a temperature of 250 K (corresponding to a total pressure and temperature of 3.8 bar and 470 K, respectively) was able to initiate a detonation in a stoichiometric ethylene-air mixture inside of a tube. However, the subsequent experimental work of Jackson and Shepherd (2004) with a design similar to that specified by Li and Kailasanath (2005) was unable to initiate ethylene-air mix- tures, even using sonic jets with total pressures and temperatures in excess (16.8 bar and 790 K) of those used in the numerical simulations (Li and Kailasanath, 2005). Simulations (Yu et al., 2004, Wang et al., 2005) have shown that the reflection of the primary explosion from the contact surface (separating the gas in the tube from
creation of high-pressures and -temperatures leading to detonation initiation in this geometry.